Permutations vs. Combinations

Both count ways to select r items from a set of n — the difference is whether order matters. A permutation treats different orderings as different outcomes (ABC ≠ BCA). A combination treats them as the same outcome (ABC = BCA, since it's the same group).

The Formulas

Both use factorials — n! means n × (n−1) × (n−2) × ... × 1. Combinations divide by an extra r! compared to permutations, since that's exactly how many different orderings of the same r items get collapsed into one combination.

How to Use This Calculator

Choose Permutation if the order of your selection matters (like assigning 1st, 2nd, and 3rd place), or Combination if it doesn't (like picking a 3-person committee where roles aren't distinguished). Enter n (your total pool) and r (how many you're selecting), then calculate.

When to Use Permutations

• Assigning distinct positions or ranks (1st/2nd/3rd place, seating order)
• Creating passwords or codes where digit order matters
• Arranging items on a shelf where position is distinguishable

When to Use Combinations

• Choosing a committee or team where roles aren't assigned
• Lottery numbers (the order you draw them doesn't change your winnings)
• Selecting a subset of items where only the group composition matters

Worked Example

Choosing 3 winners from 10 entrants for 1st, 2nd, and 3rd place (order matters — it's a permutation): 10P3 = 10!/(10−3)! = 10×9×8 = 720 possible outcomes. Choosing the same 3 people just as "the top 3" with no ranking (a combination): 10C3 = 720/3! = 120 possible groups.

Where This Fits

For broader probability work, pair this with our percentage calculator and ratio calculator.

Frequently Asked Questions